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Properties

Label 1-569-569.400-r0-0-0
Degree $1$
Conductor $569$
Sign $-0.0468 - 0.998i$
Analytic cond. $2.64242$
Root an. cond. $2.64242$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0663 + 0.997i)2-s + (−0.666 − 0.745i)3-s + (−0.991 + 0.132i)4-s + (−0.787 − 0.616i)5-s + (0.699 − 0.714i)6-s + (−0.283 − 0.958i)7-s + (−0.197 − 0.980i)8-s + (−0.110 + 0.993i)9-s + (0.562 − 0.826i)10-s + (0.903 + 0.428i)11-s + (0.759 + 0.650i)12-s + (0.862 + 0.506i)13-s + (0.937 − 0.346i)14-s + (0.0663 + 0.997i)15-s + (0.964 − 0.262i)16-s + (0.996 + 0.0883i)17-s + ⋯
L(s)  = 1  + (0.0663 + 0.997i)2-s + (−0.666 − 0.745i)3-s + (−0.991 + 0.132i)4-s + (−0.787 − 0.616i)5-s + (0.699 − 0.714i)6-s + (−0.283 − 0.958i)7-s + (−0.197 − 0.980i)8-s + (−0.110 + 0.993i)9-s + (0.562 − 0.826i)10-s + (0.903 + 0.428i)11-s + (0.759 + 0.650i)12-s + (0.862 + 0.506i)13-s + (0.937 − 0.346i)14-s + (0.0663 + 0.997i)15-s + (0.964 − 0.262i)16-s + (0.996 + 0.0883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0468 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0468 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $-0.0468 - 0.998i$
Analytic conductor: \(2.64242\)
Root analytic conductor: \(2.64242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (0:\ ),\ -0.0468 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3939073397 - 0.4128239097i\)
\(L(\frac12)\) \(\approx\) \(0.3939073397 - 0.4128239097i\)
\(L(1)\) \(\approx\) \(0.6599169907 + 0.02739264030i\)
\(L(1)\) \(\approx\) \(0.6599169907 + 0.02739264030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (0.0663 + 0.997i)T \)
3 \( 1 + (-0.666 - 0.745i)T \)
5 \( 1 + (-0.787 - 0.616i)T \)
7 \( 1 + (-0.283 - 0.958i)T \)
11 \( 1 + (0.903 + 0.428i)T \)
13 \( 1 + (0.862 + 0.506i)T \)
17 \( 1 + (0.996 + 0.0883i)T \)
19 \( 1 + (-0.991 - 0.132i)T \)
23 \( 1 + (-0.525 - 0.850i)T \)
29 \( 1 + (0.996 - 0.0883i)T \)
31 \( 1 + (-0.921 - 0.387i)T \)
37 \( 1 + (-0.283 - 0.958i)T \)
41 \( 1 + (-0.525 - 0.850i)T \)
43 \( 1 + (0.0663 - 0.997i)T \)
47 \( 1 + (-0.999 + 0.0442i)T \)
53 \( 1 + (0.699 - 0.714i)T \)
59 \( 1 + (-0.999 + 0.0442i)T \)
61 \( 1 + (0.408 - 0.912i)T \)
67 \( 1 + (0.996 - 0.0883i)T \)
71 \( 1 + (-0.197 + 0.980i)T \)
73 \( 1 + (-0.991 - 0.132i)T \)
79 \( 1 + (-0.0221 + 0.999i)T \)
83 \( 1 + (-0.839 + 0.544i)T \)
89 \( 1 + (-0.921 + 0.387i)T \)
97 \( 1 + (0.240 + 0.970i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.14202110154223591489885299229, −22.539748519252688920921884620663, −21.68440628579553071813757547984, −21.33019693168552285908653694568, −20.10570154565463442414461119018, −19.36961073579627768564902764450, −18.530032118447518088113049908756, −17.91426415726500060940632590677, −16.75908833776676735781314328308, −15.85690454441102676476952768689, −14.97558560295795973147737591090, −14.320483529954775914175696755758, −12.93395284544318185714018141737, −11.91830472609297075018086971053, −11.63587431608953596378487597534, −10.68617934963513227488780511159, −9.95023835275997752143652262936, −8.929739406170885059389634821101, −8.17001949963390858992553282768, −6.41264640766513739026588712670, −5.714024707746978511458455328986, −4.54007794571538931023542859103, −3.51420003260574719542469408807, −3.07937592890318655272977511030, −1.30320347625267464527745636557, 0.37722489881642839321032861122, 1.47496690207316943394191489004, 3.79161158260528438221488262067, 4.320003018201348599621482026898, 5.47597266839290455096242352347, 6.57453712637881203881992988010, 7.0546976191850876554541612049, 8.053474644617837212386812794579, 8.75970796893705894279264841331, 10.05081292007907692250271212676, 11.14759416193591945546419727731, 12.2898799404621457313330376603, 12.74988425107340471897210572007, 13.790779302701672650389517840380, 14.47945826298542537382425785627, 15.74693153119249592625469519420, 16.59150903501045220591717261980, 16.86551719352913831505375783595, 17.77158535982287394304078423195, 18.85537014851216442287020095382, 19.4082786276860479686852794183, 20.42490200029196192195568744284, 21.66785821235301835151250702043, 22.831275471982729321996337154, 23.17751896904830981996474100885

Graph of the $Z$-function along the critical line